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  <section id="functions">
<span id="id1"></span><h1>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h1>
<span class="target" id="module-sympy.functions"></span><p>All functions support the methods documented below, inherited from
<a class="reference internal" href="../core.html#sympy.core.function.Function" title="sympy.core.function.Function"><code class="xref py py-class docutils literal notranslate"><span class="pre">sympy.core.function.Function</span></code></a>.</p>
<dl class="py class">
<dt class="sig sig-object py">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.core.function.</span></span><span class="sig-name descname"><span class="pre">Function</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">args</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/core/function.py#L359-L828"><span class="viewcode-link"><span class="pre">[source]</span></span></a></dt>
<dd><p>Base class for applied mathematical functions.</p>
<p>It also serves as a constructor for undefined function classes.</p>
<p class="rubric">Examples</p>
<p>First example shows how to use Function as a constructor for undefined
function classes:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s1">&#39;f&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s1">&#39;g&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
<span class="go">f</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">f(x)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span>
<span class="go">g(x)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">Derivative(f(x), x)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">Derivative(g(x), x)</span>
</pre></div>
</div>
<p>Assumptions can be passed to Function, and if function is initialized with a
Symbol, the function inherits the name and assumptions associated with the Symbol:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">f_real</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s1">&#39;f&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f_real</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f_real_inherit</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="s1">&#39;f&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">f_real_inherit</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
<span class="go">True</span>
</pre></div>
</div>
<p>Note that assumptions on a function are unrelated to the assumptions on
the variable it is called on. If you want to add a relationship, subclass
Function and define the appropriate <code class="docutils literal notranslate"><span class="pre">_eval_is_assumption</span></code> methods.</p>
<p>In the following example Function is used as a base class for
<code class="docutils literal notranslate"><span class="pre">my_func</span></code> that represents a mathematical function <em>my_func</em>. Suppose
that it is well known, that <em>my_func(0)</em> is <em>1</em> and <em>my_func</em> at infinity
goes to <em>0</em>, so we want those two simplifications to occur automatically.
Suppose also that <em>my_func(x)</em> is real exactly when <em>x</em> is real. Here is
an implementation that honours those requirements:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">sin</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">my_func</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
<span class="gp">...</span>
<span class="gp">... </span>    <span class="nd">@classmethod</span>
<span class="gp">... </span>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="bp">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="gp">... </span>        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
<span class="gp">... </span>            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
<span class="gp">... </span>            <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
<span class="gp">...</span>
<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
<span class="gp">...</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
<span class="go">0</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mf">3.54</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span> <span class="c1"># Not yet implemented for my_func.</span>
<span class="go">my_func(3.54)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
<span class="go">False</span>
</pre></div>
</div>
<p>In order for <code class="docutils literal notranslate"><span class="pre">my_func</span></code> to become useful, several other methods would
need to be implemented. See source code of some of the already
implemented functions for more complete examples.</p>
<p>Also, if the function can take more than one argument, then <code class="docutils literal notranslate"><span class="pre">nargs</span></code>
must be defined, e.g. if <code class="docutils literal notranslate"><span class="pre">my_func</span></code> can take one or two arguments
then,</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">my_func</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
<span class="gp">... </span>    <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">...</span>
<span class="go">&gt;&gt;&gt;</span>
</pre></div>
</div>
<dl class="py method">
<dt class="sig sig-object py">
<span class="sig-name descname"><span class="pre">as_base_exp</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/core/function.py#L649-L653"><span class="viewcode-link"><span class="pre">[source]</span></span></a></dt>
<dd><p>Returns the method as the 2-tuple (base, exponent).</p>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py">
<span class="sig-name descname"><span class="pre">fdiff</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">argindex</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/core/function.py#L778-L800"><span class="viewcode-link"><span class="pre">[source]</span></span></a></dt>
<dd><p>Returns the first derivative of the function.</p>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py">
<em class="property"><span class="pre">classmethod</span> </em><span class="sig-name descname"><span class="pre">is_singular</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/core/function.py#L636-L647"><span class="viewcode-link"><span class="pre">[source]</span></span></a></dt>
<dd><p>Tests whether the argument is an essential singularity
or a branch point, or the functions is non-holomorphic.</p>
</dd></dl>

</dd></dl>

<section id="contents">
<span id="functions-contents"></span><h2>Contents<a class="headerlink" href="#contents" title="Permalink to this headline">¶</a></h2>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#trigonometric-functions">Trigonometric Functions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#sin">sin</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#cos">cos</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#tan">tan</a></li>
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<li class="toctree-l2"><a class="reference internal" href="elementary.html#sinc">sinc</a></li>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#trigonometric-inverses">Trigonometric Inverses</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="elementary.html#acos">acos</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#atan">atan</a></li>
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</ul>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#sympy-functions-elementary-hyperbolic">sympy.functions.elementary.hyperbolic</a></li>
<li class="toctree-l1"><a class="reference internal" href="elementary.html#hyperbolic-functions">Hyperbolic Functions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#hyperbolicfunction">HyperbolicFunction</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#sinh">sinh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#cosh">cosh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#tanh">tanh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#coth">coth</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#sech">sech</a></li>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#hyperbolic-inverses">Hyperbolic Inverses</a><ul>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#asinh">asinh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#acosh">acosh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#atanh">atanh</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#acoth">acoth</a></li>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#sympy-functions-elementary-integers">sympy.functions.elementary.integers</a><ul>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#ceiling">ceiling</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#floor">floor</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#roundfunction">RoundFunction</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#frac">frac</a></li>
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<li class="toctree-l1"><a class="reference internal" href="elementary.html#sympy-functions-elementary-exponential">sympy.functions.elementary.exponential</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="elementary.html#identityfunction">IdentityFunction</a></li>
<li class="toctree-l2"><a class="reference internal" href="elementary.html#min">Min</a></li>
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<li class="toctree-l1"><a class="reference internal" href="combinatorial.html">Combinatorial</a><ul>
<li class="toctree-l2"><a class="reference internal" href="combinatorial.html#bell">bell</a></li>
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<li class="toctree-l2"><a class="reference internal" href="special.html#module-sympy.functions.special.mathieu_functions">Mathieu Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="special.html#module-sympy.functions.special.polynomials">Orthogonal Polynomials</a></li>
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<li class="toctree-l2"><a class="reference internal" href="special.html#tensor-functions">Tensor Functions</a></li>
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